We present our regret bounds in the following theorems. These verify the conjecture made in [1]} regarding risk measures of Gaussian bandits of the form \(af(\mu ) + b g(\sigma ^2)\) , where \((f(x),g(x),a,b) = (x,x,-1,\gamma /2)\) . Furthermore, they establish ERTS as asymptotically optimal in the context of Gaussian entropic risk bandits.
[Upper Bound]
Fix \(\xi \in (0,1)\) , \(\gamma \in (0,\infty )\) . Then the asymptotic regret of ERTS for entropic risk Gaussian MAB bandits satisfies
\(\limsup _{n \rightarrow \infty } \frac{R_n(\text{ERTS})}{\log n} \le \sum _{i \in {[K]}{{1}}} R_i {i},\) 
